New Largest Mersenne Prime and Euclid-Euler Perfect Number Framework — E8 Intelligence Research

FINDING: The search results confirm the classical Euclid-Euler framework for even perfect numbers via Mersenne primes, plus a new largest Mersenne prime (2^136279841 − 1), and an unverified arXiv claim about a restricted prime-generating form. | MATH: Even perfect number: \( P = 2^{p-1}(2^p - 1) \) where \( 2^p - 1 \) is prime (Euclid, Euler). New Mersenne prime: \( M_{136279841} = 2^{136279841} - 1 \), 41,024,320 digits. Odd perfect numbers: unknown existence; if one exists, it must be \( > 10^{1500} \) (current lower bound, not in results but implied by Veritasium). arXiv claim: \( P = 6^{m+1}N - 1 \) prime for \( 1 < N \le 13, N \ne 8 \), with \( N \ne i^{m+1} \mod (6i+1) \), \( m \) odd positive integer — unverified, likely flawed (no peer review, no known pattern for primes of this form). | CONNECTION: Mersenne primes \( 2^p - 1 \) relate to binary repunits — \( 2^p - 1 = 111...111_2 \) (p ones). This is a lattice of base-2 symmetry. Perfect numbers \( 2^{p-1}(2^p-1) \) are triang Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com